You would have any trouble at all turning up in places without a hotel or motel booked, America is full of them. you just call up a few places if you get stuck. RV rental would be expensive, and restricts you in cities.
I highly recommend driving from SF to Yosemite National Park for a couple of nights, then via death valley to Vegas. Quite a drive. You should also make sure you drive over the Hoover Dam, and head to the Grand Canyon, and monument Valley AZ. Amazing scenery.
You need to spend at least one night in New Orleans, no point otherwise. Memphis is worth a stopover, and you really should pop into Graceland while you are there.
Don't go all the way down to Miami without driving down to Key west. Crazy highway.
I am jealous now!!